Congruence topologies on the mapping class group

Abstract

Let Γ(S) be the pure mapping class group of a connected orientable surface S of negative Euler characteristic. For C a class of finite groups, letπ 1 (S) C be the pro-C completion of the fundamental group of S. The C -congruence completionΓ(S) C of Γ(S) is the profinite completion induced by the embedding Γ(S) → Out(π 1 (S) C ). In this paper, we begin a systematic study of such completions for different C . We show that the combinatorial structure of the profinite groupsΓ(S) C closely resemble that of Γ(S). A fundamental question is how C -congruence completions compare with pro-C completions. Even though, in general (e.g. for C the class of finite solvable groups),Γ(S) C is not even virtually a pro-C group, we show that, for Z/2 ∈ C , g(S) ≤ 2 and S open, there is a natural epimorphism from the C -congruence com-pletionΓ(S)(2) C of the abelian level of order 2 to its pro-C completion Γ(S)(2) C . In particular, this is an isomorphism for the class of finite groups and for the class of 2-groups. Moreover, in these two cases, the result also holds for a closed surface.

Retraction of: Linear Representations of Hyperelliptic Mapping Class Groups

“…In Section 6, we study the profinite completions of this general class of hyperelliptic mapping class groups. We prove that they satisfy the congruence subgroup property, thus extending previous results from [6] and [10], and show that they are "good" in the sense of a definition by Serre. We also compute the centralizers of multitwists and of open subgroups.…”

“…Since the image of P p ΥpSq in P ΓpS W P pSqq is centralized by the hyperelliptic involution υ, it is enough to prove that the centralizer of υ in P ΓpS W P pSqq has trivial intersection with the subgroup p Πp2gpSq `2q. The last claim can be proved by the same argument given at the end of the proof of Lemma 3.7 in [6] or, alternatively, it is an immediate consequence of Lemma 6.6 in [10]. 6.4.…”

“…From Theorem 1 in [1] and the Birman short exact sequences for procongruence mapping class groups (cf. Corollary 4.7 in [10]), it follows that the kernel of f W P identifies with the group p Πp2gpSq`2q. Hence, the theorem follows if we show that the image of P p ΥpSq Ñ P ΓpS W P pSqq has trivial intersection with the subgroup p Πp2gpSq `2q.…”

“…A positive answer is only known for gpSq ď 2 (cf. [1], for gpSq " 1, and [6], [19], [10], for gpSq " 2). These are particular cases of the more general result (cf.…”

“…These are particular cases of the more general result (cf. Theorem 1.4 in [10]): Theorem 6.1. Let pS, υ, Pq be a hyperbolic marked hyperelliptic closed surface.…”